A comparison of estimation methods for fitting Weibull, Johnson’s SB and beta functions to Pinus pinaster, Pinus radiate and Pinus sylvestris stands in northwest Spain

J. J. Gorgoso, A. Rojo, A. Cámara-Obregón, U. Diéguez-Aranda

Abstract


The purpose of this study was to compare the accuracy of the Weibull, Johnson’s SB and beta distributions, fitted with some of the most usual methods and with different fixed values for the location parameters, for describing diameter distributions in even-aged stands of Pinus pinaster, Pinus radiata and Pinus sylvestris in northwest Spain. A total of 155 permanent plots in Pinus sylvestris stands throughout Galicia, 183 plots in Pinus pinaster stands throughout Galicia and Asturias and 325 plots in Pinus radiata stands in both regions were measured to describe the diameter distributions. Parameters of the Weibull function were estimated by Moments and Maximum Likelihood approaches, those of Johnson’s SB function by Conditional Maximum Likelihood and by Knoebel and Burhart’s method, and those of the beta function with the method based on the moments of the distribution. The beta and the Johnson’s SB functions were slightly superior to Weibull function for Pinus pinaster stands; the Johnson’s SB and beta functions were more accurate in the best fits for Pinus radiata stands, and the best results of the Weibull and the Johnson’s SB functions were slightly superior to beta function for Pinus sylvestris stands. However, the three functions are suitable for this stands with an appropriate value of the location parameter and estimation of parameters method.


Full Text:

PDF

References


Álvarez-González JG. 1997. Análisis y caracterización de las distribuciones diamétricas de Pinus pinaster Ait. en Galicia. Doctoral thesis, Universidad Politécnica, Madrid.

Bailey RL, Dell TR. 1973. Quantifying Diameter Distributions with the Weibull Function. For Sci 19, 97-104.

Bain LJ, Antle CE. 1967. Estimation of parameters in the Weibull distribution. Technometrics 9, 621-627.

Borders BE, Souter RA, Bailey RL, Ware KD. 1987. Percentile based distributions characterize forest stand tables. For Sci 33, 570-576.

Cao Q. 2004. Predicting parameters of a Weibull function for modeling diameter distribution. For Sci 50(5), 682- 685.

Condés S. 1997. Simulación de parcelas arboladas con datos del II Inventario Forestal Nacional. Doctoral thesis, Universidad Politécnica, Madrid.

Dans F et al. 2005. O monte galego segundo criterios de xestión forestal sostible. Diagnóstico. Asociación Forestal de Galicia, Vigo. 450 pp.

DGCN. 2006. Tercer Inventario Forestal Nacional. Ministerio de Medio Ambiente. Madrid.

Eerikäinen K, Maltamo M. 2003. A percentile based basal area diameter distribution model for predicting the stand development of Pinus kesiya plantations in Zambia and Zimbabwe. For Ecol Manage 172, 109-124.

Ek AR, Issos JN, Bailey RL. 1975. Solving for Weibull diameter distribution parameters to obtain specified mean diameters. For Sci 21, 290-292.

Fonseca TF. 2004. Modelação do crescimento, mortalidade e distribuição diamétrica, do pinhal bravo no Vale do Tâmega. Doctoral thesis, Univ. of Trás-os-Montes e Alto Douro. Vila Real.

Fonseca TF, Marques CP, Parresol BR. 2009. Describing Maritime Pine Diameter Distributions with Johnson's SB Distribution Using a New All-Parameter Recovery Approach. For Sci 55(4), 367-373.

Gerald CF, Wheatley PO. 1989. Applied numerical analysis. (4th edition). Addison-Wesley publishing Co, Reading, Massachusetts.

PMCid:2625985

Gorgoso JJ. 2003. Caracterización de las distribuciones diamétricas de Betula alba L. en Galicia. Doctoral thesis Universidad de Santiago de Compostela, Lugo.

Gorgoso JJ, Rojo A, Afif E, Barrio M. 2008. Modelling diameter distributions of birch (Betula alba L.) and pedunculate oak (Quercus robur L.) stands in northwest Spain with the beta distribution. Invest Agrar: Sist Recur For 17(3), 271-281.

Hafley WL, Schreuder HT. 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Can J For Res 4, 481-487. http://dx.doi.org/10.1139/x77-062

Harter HL, Moore AH. 1965. Maximum-Likelihood estimation of the parameters of gamma and Weibull populations from complete and from censored samples. Technometrics 7(4), 639-643. http://dx.doi.org/10.1080/00401706.1965.10490304

Hawkins K, Hotvedt J, Cao Q, Jackson B. 1988. Using the Weibull distribution to model harvesting machine productivity. Forest Products Journal 38(4), 59-65.

Hyink DM, Moser JW. 1983. A generalized framework for projecting forest yield and stand structure using diameter distributions. For Sci 29, 85-95.

Johnson NL. 1949. Systems of frequecy curves generated by methods of translation. Biometrika 36, 149-176. Johnson NL, Kitchen JO. 1971. Some notes on tables to facilitate fitting SB curves. Biometrika 58 (1), 223-226.

http://dx.doi.org/10.1093/biomet/58.1.223

Kamziah AK, Ahmad MI, Lapongan J. 1999. Nonlinear regression approach to estimating Johnson SB parameters for diameter data. Can J For Res 29(3), 310-314. http://dx.doi.org/10.1139/x98-197

Knoebel BR, Burkhart HE. 1991. A bivariate distribution approach to modeling forest diameter distributions at two points in time. Biometrics 47, 241-253. http://dx.doi.org/10.2307/2532509

Liu C, Zhang SY, Lei Y, Newton PF, Zhang L. 2004. Evaluation of three methods for predicting diameter distributions of black spruce (Picea mariana) plantations in central Canada. Can J For Res 34, 2424-2432. http://dx.doi.org/10.1139/x04-117

Loetsch F, Zöhrer F, Haller KE. 1973. Forest inventory 2. Verlagsgesellschaft. BLV. Munich. 469 pp.

Maltamo M, Puumalainen J, Päivinen R. 1995. Comparison of beta and Weibull functions for modelling basal area diameter distribution in stands of Pinus sylvestris and Picea abies. Scan J For Res 10, 284-295. http://dx.doi.org/10.1080/02827589509382895

Maltamo M, Kangas A. 1998. Methods based on k-nearest neighbour regression in the prediction of basal area diameter distribution. Can J For Res 28, 1107-1115. http://dx.doi.org/10.1139/x98-085

Mønness EN. 1982. Diameter distributions and height curves in even aged stands of Pinus sylvestris L. Meddeleser fra Norsk Institut for Skoforskning 36(15), 1-43.

Nanang DM. 1998. Suitability of the Normal, Log-normal and Weibull distributions for fitting diameter distributions of neem plantations in Northern Ghana. For Ecol Manage 103, 1-7.

Nanos N, Montero G. 2002. Spatial prediction of diameter distributions models. For Ecol Manage 161, 147-158.

Palahí M, Pukkala T, Blasco E, Trasobares A. 2007. Comparison of beta, Johnson's SB, Weibull and truncated Weibull functions for modeling the diameter distribution of forest stands in Catalonia (north-east of Spain). Eur J Forest Res 126, 563-571. http://dx.doi.org/10.1007/s10342-007-0177-3

Parresol BR. 2003. Recovering parameters of Johnson's SB distribution. US For. Ser. Res. Paper SRS-31. 9 pp.

Río M. del 1999. Régimen de claras y modelo de producción para Pinus sylvestris L. en los sistemas Central e Ibérico. Tesis doctorales INIA, Serie: Forestal 2. Madrid, España, 257 pp.

SADEI. 2010. Datos básicos de Asturias 2010. Instituto Asturiano de Estadística. Gobierno del Principado de Asturias. 40 pp.

SAS Institute INC. 2003. SAS/STATTM User's Guide, Version 9.1. Cary, North Carolina.

Scolforo JRS, Vitti FC, Grisi RL, Acerbi F, De Assis AL. 2003. SB distribution's accuracy to represent the diameter distribution of Pinus taeda, through five fitting methods. For Ecol Manage 175, 489-496.

Shifley SR, Lentz EL. 1985. Quick estimation of the threeparameter Weibull to describe tree size distributions. For Ecol Manage 13, 195-203.

Siekierski K. 1992. Comparison and evaluation of three methods of estimation of the Johnson SB distribution. Biom J 34, 879-895.

http://dx.doi.org/10.1002/bimj.4710340714

Siipilehto J. 1999. Improving the accuracy of predicted basal area diameter distribution in advanced stands by determining stem number. Silva Fenn 33(4), 281-301.

Stankova TV, Zlatanov TM. 2010. Modeling diameter distribution of Austrian black pine (Pinus nigra Arn.) plantations: a comparison of the Weibull frequency distribution function and percentile-based projection methods. Eur J Forest Res 129, 1169–1179. http://dx.doi.org/10.1007/s10342-010-0407-y

Zanakis SH. 1979. A simulation study of some simple estimators for the three parameter Weibull distribution. J Stat Comput Simul 9, 101-116. http://dx.doi.org/10.1080/00949657908810302

Zhang L, Packard KC, Liu C. 2003. A comparison of estimation methods for fitting Weibull and Johnson's SB distributions to mixed spruce-fir stands in northeastern North America. Can J For Res 33, 1340-1347. http://dx.doi.org/10.1139/x03-054

Zhou B, McTague JP. 1996. Comparison and evaluation of five methods of estimation of the Johnson System parameters. Can J For Res 26, 928-935. http://dx.doi.org/10.1139/x26-102

Zöhrer F. 1969. The application of the beta function for best fit of stem diameter distributions in inventories of tropical forest. Mitt. Bundesforsch-.anst. Forst- u. Holzwirtsch., Reinbek/Hamburg 74: 279-293.

Zöhrer F. 1970. Das Computerprogram BETKLA zum Ausgleich von Stammzahl-Durchmesserverteilungen mit Hilfe der Beta-Verteilung. Mitt. Bundesforsch-.anst. Forst- u. Holzwirtsch., Reinbek/Hamburg 76, 50 pp.




DOI: 10.5424/fs/2012213-02736

Webpage: www.inia.es/Forestsystems